Let PG(V) and PG(W) be projective geometriesMathworldPlanetmath, with V,W vector spacesMathworldPlanetmath over a field K. A function p from PG(V) to PG(W) is called a projective transformation, or simply a projectivity if

  1. 1.

    p is a bijectionMathworldPlanetmath, and

  2. 2.

    p is order preserving.

A projective property is any geometric property, such as incidence, linearity, etc… that is preserved under a projectivity.

From the definition, we see that a projectivity p carries 0 to 0, V to W. Furthermore, it carries points to points, lines to lines, planes to planes, etc.. In short, p preserves linearity. Because p is a bijection, p also preserves dimensionsPlanetmathPlanetmath, that is dim(S)=dim(p(S)), for any subspacePlanetmathPlanetmathPlanetmath S of V. In particular, dim(V)=dim(W). Other properties preserved by p are incidence: if ST, then p(S)p(T); and cross ratios (

Every bijectiveMathworldPlanetmath semilinear transformation defines a projectiviity. To see this, let f:VW be a semilinear transformation. If S is a subspace of V, then f(S) is a subspace of W, as x,yf(S), then x+y=f(a)+f(b)=f(a+b)f(S), and αx=βθx=βθf(a)=f(βa)f(S), where θ is an automorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath of the common underlying field K. Also, if S is a subspace of a subspace T of V, then f(S) is a subspace of f(T). Now if we define f*:PG(V)PG(W) by f*(S)=f(S), it is easy to see that f* is a projectivity.

Conversely, if V and W are of finite dimension greater than 2, then a projectivity p:PG(V)PG(W) induces a semilinear transformation p^:VW. This highly non-trivial fact is the (first) fundamental theorem of projective geometryMathworldPlanetmath.

If the semilinear transformation induced by the projectivity p is in fact a linear transformation, then p is a collineationMathworldPlanetmath: three distinct collinear points are mapped to three distinct collinear points.

Remark. The definition given in this entry is a generalizationPlanetmathPlanetmath of the definition typically given for a projective transformation. In the more restictive definition, a projectivity p is defined merely as a bijection between two projective spacesMathworldPlanetmath that is induced by a linear isomorphism. More precisely, if P(V) and P(W) are projective spaces induced by the vector spaces V and W, if L:VW is a bijective linear transformation, then p=P(L):P(V)P(W) defined by


is the corresponding projective transformation. [v] is the homogeneous coordinate representation of v. In this definition, a projectiity is always a collineation. In the case where the vector spaces are finite dimensional with specified bases, p is expressible in terms of an invertible matrix (Lv=Av where A is an invertible matrix).

Title projectivity
Canonical name Projectivity
Date of creation 2013-03-22 15:58:00
Last modified on 2013-03-22 15:58:00
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 51A10
Classification msc 51A05
Related topic PolaritiesAndForms
Related topic SesquilinearFormsOverGeneralFields
Related topic PerspectivityMathworldPlanetmath
Related topic ProjectiveSpace
Related topic LinearFunction
Related topic Collineation
Defines projective transformation
Defines projective property